Mathematics Pacing Resource Document
6.NS.10
Standard: 6.NS.10: Use reasoning involving rates and ratios to model real-world and other mathematical problems (e.g., by reasoning about tables of
equivalent ratios, tape diagrams, double number line diagrams, or equations).
Teacher Background Information:
Ratios and rates can be used in ratio tables and graphs to solve problems. Previously, students have used additive reasoning in tables to solve
problems. To begin the shift to proportional reasoning, students need to begin using multiplicative reasoning. To aid in the development of
proportional reasoning the cross-product algorithm is not expected at this level. When working with ratio tables and graphs, whole number
measurements are the expectation for this standard.
Example 1:
At Books Unlimited, 3 paperback books cost $18. What would 7 books cost? How many books could be purchased with $54.
Solution: To find the price of 1 book, divide $18 by 3. One book costs $6. To find the price of 7 books, multiply
$6 (the cost of one book times 7 to get $42. To find the number of books that can be purchased with $54, multiply
$6 times 9 to get $54 and then multiply 1 book times 9 to get 9 books. Students use ratios, unit rates and multiplicative reasoning to solve problems
in various contexts, including measurement, prices, and geometry. Notice in the table below, a multiplicative relationship exists between the
numbers both horizontally (times 6) and vertically (ie. 1 • 7 = 7; 6 • 7 = 42). Red numbers indicate solutions.
Number
of Books
(n)
1
3
Cost
(C)
7
9
42
54
6
18
Students use tables to compare ratios. Another bookstore offers paperback books at the prices below. Which bookstore has the best buy? Explain
your answer.
Indianapolis Public Schools
Curriculum and Instruction
Mathematics Pacing Resource Document
6.NS.10
Standard: 6.NS.10: Use reasoning involving rates and ratios to model real-world and other mathematical problems (e.g., by reasoning about tables of
equivalent ratios, tape diagrams, double number line diagrams, or equations).
Teacher Background Information:
Number
of Books
(n)
Cost
(C)
4
20
8
40
To help understand the multiplicative relationship between the number of books and cost, students write equations to express
the cost of any number of books. Writing equations is foundational for work in 7th grade. For example, the equation for the
first table would be C = 6n, while the equation for the second bookstore is C = 5n.
The numbers in the table can be expressed as ordered pairs (number of books, cost) and plotted on a coordinate plane.
Students are able to plot ratios as ordered pairs. For example, a graph of Books Unlimited would be:
Indianapolis Public Schools
Curriculum and Instruction
Mathematics Pacing Resource Document
6.NS.10
Standard: 6.NS.10: Use reasoning involving rates and ratios to model real-world and other mathematical problems (e.g., by reasoning about tables of
equivalent ratios, tape diagrams, double number line diagrams, or equations).
Teacher Background Information:
Example 2:
Ratios can also be used in problem solving by thinking about the total amount for each ratio unit.
The ratio of cups of orange juice concentrate to cups of water in punch is 1: 3. If James made 32 cups of punch, how many cups of orange did he
need?
Solution: Students recognize that the total ratio would produce 4 cups of punch. To get 32 cups, the ratio would need to be duplicated 8 times,
resulting in 8 cups of orange juice concentrate.
Example 3:
Using the information in the table, find the number of yards in 24 feet.
Solution:
There are several strategies that students could use to determine the solution to this problem:
o Add quantities from the table to total 24 feet (9 feet and 15 feet); therefore the number of yards in 24 feet must be 8 yards (3 yards and 5
yards).
o Use multiplication to find 24 feet: 1) 3 feet x 8 = 24 feet; therefore 1 yard x 8 = 8 yards, or 2) 6 feet x
4 = 24 feet; therefore 2 yards x 4 = 8 yards.
Indianapolis Public Schools
Curriculum and Instruction
Mathematics Pacing Resource Document
6.NS.10
Standard: 6.NS.10: Use reasoning involving rates and ratios to model real-world and other mathematical problems (e.g., by reasoning about tables of
equivalent ratios, tape diagrams, double number line diagrams, or equations).
Teacher Background Information:
Example 4:
Compare the number of black circles to white circles. If the ratio remains the same, how many black circles will there be if there are 60 white circles?
Black
4
40
20
60
?
White
3
30
15
45
60
Solution:
There are several strategies that students could use to determine the solution to this problem
o Add quantities from the table to total 60 white circles (15 + 45).
Use the corresponding numbers to determine the number of black circles (20 + 60) to get 80 black circles.
o Use multiplication to find 60 white circles (one possibility 30 x 2).
Use the corresponding numbers and operations to determine the number of black circles (40 x 2) to get 80 black circles.
Students recognize the use of ratios, unit rate and multiplication in solving problems, which could allow for the use of fractions and decimals.
Example 5:
In trail mix, the ratio of cups of peanuts to cups of chocolate candies is 3 to 2. How many cups of chocolate candies would be needed for 9 cups of
peanuts.
Peanuts Chocolate
3
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2
Curriculum and Instruction
Mathematics Pacing Resource Document
6.NS.10
Standard: 6.NS.10: Use reasoning involving rates and ratios to model real-world and other mathematical problems (e.g., by reasoning about tables of
equivalent ratios, tape diagrams, double number line diagrams, or equations).
Teacher Background Information:
Solution:
One possible solution is for students to find the number of cups of chocolate candies for 1 cup of peanuts by dividing both sides of the table by 3,
2
giving cup of chocolate for each cup of peanuts. To find the amount of chocolate needed for 9 cups of peanuts, students multiply the unit rate by
3
2
nine (9 x3) giving 6 cups of chocolate.
Example 6:
If steak costs $2.25 per pound, how much does 0.8 pounds of steak cost? Explain how you determined your answer.
Solution:
The unit rate is $2.25 per pound so multiply $2.25 x 0.8 to get $1.80 per 0.8 lb. of steak.
Example 7:
The Business Direct Hotel caters to people who travel for different types of business trips. On Saturday night there is not a lot of business travel, so the ratio of
the number of occupied rooms to the number of unoccupied rooms is 2:5. However, on Sunday night the ratio of the number of occupied rooms to the number
of unoccupied rooms is 6:1 due to the number of business people attending a large conference in the area. If the Business Direct Hotel has 432 occupied rooms
on Sunday night, how many unoccupied rooms does it have on Saturday night?
http://www.btboces.org/Downloads/04introduction_to_tape_diagrams.pdf Introduction to Tape
Diagrams
http://www.lpssonline.com/uploads/introduction_to_doub
le_number_lines.pdf Introduction to Double Number Line
Diagrams
Process Standards to Emphasize with Instruction of 6.NS.10:
6.PS.1: Make sense of problems and persevere in solving them.
6.PS.2: Reason abstractly and quantitatively.
6.PS.4: Model with mathematics.
6.PS.5: Use appropriate tools strategically.
6.PS.7: Look for and make use of structure.
Indianapolis Public Schools
Curriculum and Instruction
Mathematics Pacing Resource Document
6.NS.10
Standard: 6.NS.10: Use reasoning involving rates and ratios to model real-world and other mathematical problems (e.g., by reasoning about tables of
equivalent ratios, tape diagrams, double number line diagrams, or equations).
Lesson Plans/Print Activities:
Web-based Practice:
 Indiana Math Connects: Lesson 6-2
 IMPACT Math: Unit D Investigation 4
 http://glencoe.mheducation.com/sites/0078923379/student_view0/ch
apter6/lesson3/personal_tutor.html Glencoe Personal Tutor
 https://www.engageny.org/resource/grade-6-mathematics-module-1-topic-boverview EngageNY Module 1 Topic B Lessons 9-15: Students explore collections of
equivalent ratios in real world contexts.
 http://glencoe.mheducation.com/sites/0078923379/student_view0/ch
apter6/lesson4/group_activity_cards.html Group Activity Card
 https://www.engageny.org/resource/grade-6-mathematics-module-1-topic-c-lesson19 EngageNY Module 1 Topic C lessons 19-23: Students solve problems by analyzing
different unit rates given in tables, equations, and graphs.
 http://www.thinkingblocks.com/tb_ratios/ratios_mp.html Math
Playground Thinking Blocks
 https://www.illustrativemathematics.org/illustrations/135 Illustrative Mathematics
Task: Painting a Barn
 https://www.illustrativemathematics.org/illustrations/498 Illustrative Mathematics
Task: Jim and Jesse’s Money
 http://www.bsquaredfutures.com/pluginfile.php/137/mod_resource/c
ontent/2/swf/highHitter.swf Tower of Strength
 https://www.khanacademy.org/commoncore/grade-6-RP Khan
Academy Ratios and Proportional Relationships (Scroll down to
6.RP.A.3)
 http://illuminations.nctm.org/Lesson.aspx?id=3128 Illuminations: Highway Robbery
 http://schools.nyc.gov/NR/rdonlyres/A9F735CB-47E4-40F8-884FEA54D0AB5705/0/NYCDOEG6MathRatios_FINAL.pdf NYC.gov Ratios and Proportional
Relationships Unit
 http://www.learningfarm.com/web/practicePassThrough.cfm?TopicID=
1299 Learning Farm Rates and Ratios
 http://www.cpalms.org/Public/PreviewResource/Preview/47977 This lesson shows
how ratios can be indicated in words such as "to", "for every", "out of every." In grade
6, rates mean "for each 1," "for each," and "per."
 https://learnzillion.com/resources/64213-grade-6-unit-2 LearnZillion Rates Unit. Scroll
down for lesson descriptors.
 https://learnzillion.com/resources/74923-6th-grade-ratios-and-proportionalrelationships LearnZillion video lessons
Indianapolis Public Schools
Curriculum and Instruction
Mathematics Pacing Resource Document
6.NS.10
Item Bank:
Use the table below to answer questions 1-2
Asha found that 6 cars passed her house in 5 minutes. Using this rate, she made a partially complete ratio table to show the relationship between time and
number of cars.
Cars
6
12
18
X
Minutes
5
10
15
20
1) At this rate, how many cars will Asha expect to see over 1 hour? (Answer: B)
A) 24 cars
B) 72 cars
C) 25 cars
D) 21 cars
2) At this rate, how many cars should Asha expect to see in 2 ½ minutes? (Answer: C)
A) 12 cars
B) 2.5 cars
C) 3 cars
D) 25 cars
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Mathematics Pacing Resource Document
6.NS.10
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(Readiness I)
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(Readiness II)
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Item Bank:
Use the information below to answer questions 3 – 5
Zumba is a great dance/cardio workout. The following table shows the relationship between the time spent doing Zumba and how many calories are burned.
ZUMBA calories burned
Time
(minutes)
Calories
burned
10
X
20
107.2
30
160.8
3) What is the unit rate (per minute) of calories burned? Answer:5.36 calories per minute
Answer:_______________________
4) What is x = to in the table? Answer: B
A) 5.36
B) 53.6 C) 32.3 D) 188.2
5) How many calories could you burn in 2 hours of Zumba? Answer: B
A) 321.6 B) 643.2 C) 160.8 D) 262.34
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6.NS.10
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(SBAC)
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Mathematics Pacing Resource Document
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